MinT - Best Known (171−89, 171, s)-Nets in Base 4

Best Known (171−89, 171, s)-Nets in Base 4

(171−89, 171, 104)-Net over F₄ — Constructive and digital

Digital (82, 171, 104)-net over F₄, using

t-expansion [i] based on digital (73, 171, 104)-net over F₄, using
- net from sequence [i] based on digital (73, 103)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 73 and N(F) ≥ 104, using
    - F₆ from the tower of function fields by GarcÃa and Stichtenoth over F₄ [i]

(171−89, 171, 129)-Net over F₄ — Digital

Digital (82, 171, 129)-net over F₄, using

t-expansion [i] based on digital (81, 171, 129)-net over F₄, using
- net from sequence [i] based on digital (81, 128)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 81 and N(F) ≥ 129, using
    - Drinfeld modules of rank 1 [i]

(171−89, 171, 1183)-Net in Base 4 — Upper bound on s

There is no (82, 171, 1184)-net in base 4, because

1 times m-reduction [i] would yield (82, 170, 1184)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 2 316566 252270 373977 917667 838203 292369 757091 326361 287487 360723 146444 705733 388617 797433 153812 671059 957620 > 4¹⁷⁰ [i]